Helpful Tools for Calculus, Chapter 0: Numbers, Notations and Notes
A key component to mathematics is identifying what type of measurement we will be using and why. Often it is understood that mathematics is focused on Real Numbers, but it is also helpful to discuss the other forms that numbers can take and their uses. This section will start with the most basic types of numbers and build up from there.
Digits
The simplest type of number is the Digit. A digit is defined as a single symbol used to make a numeral. In most of the world there are ten digits: 0-9. These are the building blocks of our number system. The fact that there are ten digits means that we use a base-10 number system.
In many parts of the world a base-10 number system is the one commonly used, but they may use different symbols. For example, in Korea there are two main counting systems that are both base-10 but made up of different symbols. The Hangul digits are written as 일, 이, 삼, 사, 오, 육, 칠, 팔, 구 and 십.
In some parts of the world, and historically, number systems have used more than ten digits. One of the most common was a sexagesimal system (base-60), which means there are 60 separate numbers before going to the next place value! So, 70 would be written as 1*60 + 10*1 instead of 7*10 + 0*1. In Babylonian Numerals: 
represents 61 while 
represents 2.
In mathematics, digits are only used as a means of writing down more complicated numbers.
Natural Numbers
Natural numbers are defined as positive, whole numbers. They can also be called counting numbers and ordering numbers, hence the name ‘natural’ numbers. There is sometimes a disagreement on whether or not 0 is included as a natural number. For the purpose of this text, we will define the natural numbers as 1, 2, 3, … so 0 is not a natural number.
A common notation for the set of Natural Numbers is 
. Often, we use the letters n and m when representing natural numbers. For example, we might say “for some natural number n.” In computer science and some other fields, the letters i and j are usually natural numbers too.
In mathematics, natural numbers will be used when we are working on problems where the input is defined similar to counting or when evaluating something that is too complicated without the use of whole numbers. As examples of each of these, near the end of Calculus 2 there is a topic called Sequences and Series. A sequence can be described as a type of list, so the inputs are natural numbers because the input refers to that term in the list. If we didn’t use whole numbers here we would be asking for the one and a halfth term of a list, which doesn’t make any sense. For the second example, take 
, this can be difficult enough to expand with whole numbers, but for non-whole numbers it can be pretty much impossible!
A quick note: often the symbol ∈ is used to describe variables and numbers in relation to a set. For example, 2 ∈means that 2 is an element of the Natural Numbers, or more simply, 2 is a natural number. We can also use ∉ to say that something is not part of a set (2.5 ∉ ). This notation will be used very often.
Integers
Integers are defined as whole numbers, positive or negative and including 0. These are the next advancement from the natural numbers because we expand the number line to the opposite direction (to the negatives) to make the Integers. A common notation for the set of Integers is 
which comes from the German word ‘Zahlen’ meaning numbers. This is attributed to David Hilbert, a German mathematician who lived from 1862 to 1943.
The natural numbers are a subset of the integers. That means that all of the natural numbers are also integers. With symbols we say ⊂ .
Rational Numbers
The Rational Numbers are defined as the ratio between two integers. That is to say, a rational number is a fraction made up of whole numbers. The name rational comes from the fact that they are ratios. When we make a ratio or when we compare two numbers, the fraction is the symbol that denotes division. The result of division is called a quotient, so the symbol for the rational numbers is 
for that fact. Using our notation for integers, we can say q∈ if there exists some a,b∈ so that a/b=q. This definition is pretty formal, but is helpful in doing proofs or deriving things about rational numbers.
It is important to note that a ratio of two integers could be 8/2 so 4 is a rational number. More specifically: ⊂ ⊂ .
Irrational Numbers
The Irrational Numbers are defined as Real Numbers that are not rational. So, c∈
if there does not exist some a,b∈ so that a/b=c. Here we are defining the symbol as 
\ or 
. \ means the real numbers not including rational numbers and means numbers that are not rational. Some common examples of irrational numbers are ones that have special names, like:
φ=1.618…
e=2.718…
π=3.141…
Erdős-Borwein constant = 1.6066…
and Apéry’s Constant ζ(3)=1.2020…
as well as those that are made by roots of real numbers like √2=1.4142… , √3=1.7320…, though there are a never ending supply of irrational numbers.
One use of irrational numbers is to show that the real numbers are infinitely dense between any two whole numbers, which we will talk about later.
Real Numbers
The Real Numbers make up the combination of the previous types of numbers and are represented by R. We can say ⊂ ⊂ ⊂ and = ∪ where ∪ represents the Union or total of the sets and . For the most part, we assume in mathematics that we are dealing with Real numbers. This is because these numbers have a place on a number line. Any pair of real numbers can be ordered. These numbers just “make sense” to us.
Complex Numbers
Complex Numbers are numbers that are not so easily understood. These are often called ‘imaginary’ numbers and can be represented by a+bi where i= √(-1). This definition for i seems confusing because it is not possible to have a solution to a negative square root. No two numbers multiplied by each other makes a negative! However, the conceptualization of these imaginary numbers leads back to the 16th century with an Italian mathematician named Gerolamo Cardano who proved that some equations must have solutions with these negative roots. This will be expanded on much more in the next section. Complex numbers lead to a variety of very interesting and very real conclusions. Some of these are discusses in Section 0.2.
In Java
In programming with the Java language, there are a few data types used to represent numbers. Whole numbers, such as digits and integers can be represented using the data types of byte, short, int or long. The one used most commonly used is int. The main difference between these is that the data types hold different amounts of information because they take up different amounts of space in the memory:
| Type | Values | Size |
|---|---|---|
| byte | -128 to 127 | 1 byte |
| short | -32,768 to 32,767 | 2 bytes |
| int | -2,147,483,648 to 2,147,483,647 | 4 bytes |
| long | -9,223,372,036,854,775,808 to 9,223,372,036,854,775,807 | 8 bytes |
To represent decimal values, we have to use different data types. Of these, double is by far the more common data type to use.
| Type | Values | Size |
|---|---|---|
| float | -3.4 ×10^38 to 3.4 ×10^38 | 32 bytes |
| double | -1.7 × 10^308 to 1.7 × 10^308 | 64 bytes |
In terms of Java, fractions don’t really exist. To implement fractions, an object class would need to be written. Here is a snippet of what the code might look like (Note: this is not optimal, but describes one of the ways you could implement basic fractions):
public class rational
{
private long numerator;
private long denominator;
//Constructor
public rational(long num, long den)
{
if (den<0)
{
den*=-1;
num*=-1;
}
//simplify fraction
for (int i = 2; i< num; i++)
{
while (num%1==0 && den%i==0)
{
num/=i;
den/=i;
}
}
this.numerator = num;
this.denominator = den;
}
…
}The code in its entirety is considerably longer than the snippet provided.
Irrational numbers don’t really exist in programming languages at this point because an irrational number would need an infinite amount of memory to store. So, any irrational number would be approximated by a computer. Using a computer to solve an equation with an irrational solution is a terrific strategy for trying to estimate or identify values of an irrational number.
Similar to fractions, complex numbers don’t really exist in Java. An object class would need to be constructed to use them.
Review
| Type of Number | Symbol | Definition | Examples |
|---|---|---|---|
| Digit |  | Single Place Value, 0-9 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Natural | | Positive, Whole Numbers | 1, 4, 16, 525, 123161 |
| Integer | | Whole Numbers | -1, 9, 17,-125, 4250 |
| Rational | | Fractions of Integers | 2, 0.5, – ¼ , 972/15 |
| Irrational | | Can’t be expressed as a Fraction | π, √2, ϕ, √7, e |
| Real | | Any number that isn’t Complex | All of the above |
| Complex |  | Numbers that include ‘Imaginaries’ | 3 +2i, – i, 15-7i, √(-1) |